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Young Won Lim6/23/14
Derivatives (1A)
Young Won Lim6/23/14
Copyright (c) 2011 - 2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
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Derivatives (1A) 3 Young Won Lim6/23/14
A triangle and its slope
y = f x
f x1 h − f x1h
x1, f x1
x1 h , f x1 h
x1, f x1
http://en.wikipedia.org/wiki/Derivative
x1 x1+h
f (x1+h)
f (x1)
Derivatives (1A) 4 Young Won Lim6/23/14
Many smaller triangles and their slopes
f ( x1)
(x1, f (x1))
f ( x1+h2)
f ( x1+h1)
f ( x1+h)
f (x1 + h) − f ( x1)h
f (x1 + h1) − f (x1)h1
f (x1 + h2) − f (x1)h2
limh→0
f (x1 + h) − f (x1)h
x1 x2+h2 x2+h1 x2+h
(x1+h , f (x1+h))
h2h1
h
(x1, f (x1))
(x1+h , f (x1+h))
http://en.wikipedia.org/wiki/Derivative
Derivatives (1A) 5 Young Won Lim6/23/14
The limit of triangles and their slopes
y = f x
f ' (x1) = limh→0
f (x1 + h) − f (x1)h
x1
f
The derivative of the function f at x1
function f
http://en.wikipedia.org/wiki/Derivative
Derivatives (1A) 6 Young Won Lim6/23/14
The derivative as a function
y = f x x1x2x3
f x1
f x2
f x3
f x
x1x2x3
f ' x 1
f ' x 2
f ' x 3
f ' x
= limh→0
f (x + h) − f (x)h
y ' = f ' (x)
Derivative Function
Derivatives (1A) 7 Young Won Lim6/23/14
The notations of derivative functions
x1x2x3
f ' x 1
f ' x 2
f ' x 3
f ' (x)
limh→0
f (x + h) − f (x )h
y ' = f ' x Largrange's Notation
dydx
=ddx
f x
Leibniz's Notation
ẏ = ḟ x Newton's Notation
D x y = D x f x Euler's Notation
not a ratio.
x is an independent variable→ Derivative with respect to x
Derivatives (1A) 8 Young Won Lim6/23/14
Another kind of triangles and their slopes
y = f x
f
y ' = f ' x given x1
the slope of a tangent line
http://en.wikipedia.org/wiki/Derivative
Derivatives (1A) 9 Young Won Lim6/23/14
Differential in calculus
x1 x1 dx
f x1
f x1 dx
dx
dy
Differential: dx, dy, … ● infinitesimals● a change in the linearization of a function● of, or relating to differentiation
the linearization of a function
function
http://en.wikipedia.org/wiki/Derivative
Derivatives (1A) 10 Young Won Lim6/23/14
Differential as a function
x1
f x1
Line equation in the new coordinate.
dx
dydy = f ' x1 dx
slope = f ' x1
http://en.wikipedia.org/wiki/Derivative
Derivatives (1A) 11 Young Won Lim6/23/14
Differentiation & Integration of sinusoidal functions
f x = sin x dd x
f x = cos x leads
f x = cos xdd x
f x = −sin x leads
f x = sin x ∫ f x dx = −cos x C lags
f x = cos x∫ f x dx = sin x C lags
Derivatives (1A) 12 Young Won Lim6/23/14
Derivative of sin(x)
f (x) = sin(x)
+1 0 -1 0 +1 0 -1 0 +1 0 -1 0 slope
dd x
f (x ) = cos(x )
leads
Derivatives (1A) 13 Young Won Lim6/23/14
Plot of F(x,y)=cos(x)
+1 0 -1 0 +1 0 -1 0 +1 0 -1 0 slope
F (x , f (x )) = f ' (x) (x i , y i) f ' (x i)
slope=+1
F (x , y)
+1
0
-1
0
+1
0
-1
0
+1
0
-1
0
f (x ) = sin(x)
f ' (x ) = cos(x)
Derivatives (1A) 14 Young Won Lim6/23/14
Derivative of cos(x)
f x = cos x
0 -1 0 +1 0 -1 0 +1 0 -1 0 +1 slope
dd x
f x = −sin x
leads
Derivatives (1A) 15 Young Won Lim6/23/14
Plot of F(x,y)=-sin(x)
F (x , f (x )) = f ' (x) (x i , y i) f ' (x i)
slope=+1
F (x , y)
+1
-1
+1
-1
+1
-1
0 -1 0 +1 0 -1 0 +1 0 -1 0 +1 slope
0 0 0 0 0 0
f (x ) = cos(x)
f ' (x ) = −sin (x)
Derivatives (1A) 16 Young Won Lim6/23/14
Integral of sin(x)
f x = sin x
0 1 2 1 0 1 2 1 0 1 2 1 area + 0
∫ f x dx = −cos x C
-1 0 +1 0 -1 0 +1 0 -1 0 +1 0 area - 1
∫0/2
sin t d t = 1
C = 0
C = 1 ∫0xsin (t) d t + 0
∫0xsin (t) d t − 1
= − cos x lags
Derivatives (1A) 17 Young Won Lim6/23/14
Integral of cos(x)
f x = cos x
0 1 0 -1 0 1 0 -1 0 1 0 -1 area
∫ f x dx = sin x C
∫0/2
cos x d x = 1
lags
∫0xcost d t
= + sin(x)
Derivatives (1A) 18 Young Won Lim6/23/14
The Euler constant e
dd x
ax = limh0
a xh − ax
h= a x lim
h0
ah − 1h
limh→0
ah − 1h
= 1 f ' 0 = 1
dd x
ex = e x
f x = e x f ' x = ex f ' ' x = ex
limh→0
ah − a0
h − 0= 1
e = 2.71828⋯
⇒ a x such a, we call e
Derivatives (1A) 19 Young Won Lim6/23/14
The Euler constant e
f ' 0 = 1
dd x
ex = e x
f x = e x
f ' x = ex
f ' ' x = ex
limh→0
ah − a0
h − 0= 1
e = 2.71828⋯
limh→0
ah − 1h
= 1 iif a = e
Functions f(x) = ax are shown for several values of a. e is the unique value of a, such that the derivative of f(x) = ax at the point x = 0 is equal to 1. The blue curve illustrates this case, ex. For comparison, functions 2x (dotted curve) and 4x (dashed curve) are shown; they are not tangent to the line of slope 1 and y-intercept 1 (red).
http://en.wikipedia.org/wiki/Derivative
Derivatives (1A) 20 Young Won Lim6/23/14
Derivative Product and Quotient Rule
f (x)g(x ) f ' (x)g(x) + f ( x)g ' (x)dd x
ddx
( f g) = d fdxg + f d g
dx
f (x )g( x)
dd x
ddx ( fg ) = ( d fdx g − f d gdx ) / g2
f ' (x )g(x ) − f (x)g ' (x )g2(x )
Derivatives (1A) 21 Young Won Lim6/23/14
Integration by parts (1)
f (x)g(x ) f ' (x)g(x) + f ( x)g ' (x)dd x
ddx
( f g) = d fdxg + f d g
dx
f (x)g(x ) f ' (x)g(x) + f ( x)g ' (x)
f g = ∫ f ' g dx + ∫ f g ' dx
∫⋅d x
∫ f (x )g ' (x) dx f (x )g(x ) − ∫ f ' (x )g(x ) dx=
Derivatives (1A) 22 Young Won Lim6/23/14
Derivative of Inverse Functions
y = f (x )
a
b (a, b) = (a , f (a))
b
a(b, a) = (b , f−1(b))
y = f−1(x )= g(x )
g' (b) = 1f ' (a)
m2 = g' (b)m1 = f ' (a)
= 1f ' (g(b))
= (b ,g(b))
m1m2 = f ' (a)g ' (b)= 1
Derivatives (1A) 23 Young Won Lim6/23/14
Derivative of ln(x) (1)
y = f (x )
a
b (a, b) = (a , f (a))
b
a(b, a) = (b , f−1(b))
y = f−1(x )= g(x )
g' (b) = 1f ' (a) m = g' (b)m = f ' (a )
= 1f ' (g(b))
= (b ,g(b))
f (x )= ex g(x ) = ln x = f −1(x)
g' (x )= 1f ' (g(x))
= 1eg(x )
= 1e ln x
= 1x
Derivatives (1A) 24 Young Won Lim6/23/14
Derivative of ln |x|
y = ln x
e y = x
ddx
e y = ddx
x
e y dydx
= 1
dydx
= 1e y
= 1x
ddx
ln x = 1x
x>0
ddx
ln(−x )= 1(−x )
⋅(−1) = 1x
x
Derivatives (1A) 25 Young Won Lim6/23/14
Differential Equation
f (x ) = e3 x
f ' (x ) = 3e3 x
f ' (x ) − 3 f (x) = 3 e3x−3 e3 x = 0 f ' (x ) − 3 f (x) = 0
f (x ) ?
Young Won Lim6/23/14
References
[1] http://en.wikipedia.org/[2] M.L. Boas, “Mathematical Methods in the Physical Sciences”[3] E. Kreyszig, “Advanced Engineering Mathematics”[4] D. G. Zill, W. S. Wright, “Advanced Engineering Mathematics”
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